To see the connection between frustrated spins in the system consider \(e=|E(\mathcal {G})|\)  the total number of edges (ordered pairs of adjacent vertices) of the graph [1]}, while \(n=|V(\mathcal {G})|\)  is the total number of nodes. Let us first introduce a few graph theoretical constructs in order to fix the notation.
For a generic and undirected graph \(\mathcal {G}\)  consider the (undirected) incidence matrix \(B\)  of size \(n \times e\)  with entries \(B_{i\beta }\) , where \(i\)  is an integer between 1 and \(n\)  on the set of vertices and \(\beta \)  is an integer between 1 and \(e\)  on
the set of edges, such that:
\(B_{i\beta } = \left\lbrace \begin{array}{rl}1 & \text{if the edge $\beta $ contains the vertex $i$} ,\\0 & \text{otherwise }.\end{array} \right.\) 
